partial ordering

A relation R is a partial ordering if it is a pre-order
(i.e. it is reflexive (x R x) and transitive (x R y R z =>
x R z)) and it is also antisymmetric (x R y R x => x = y).
The ordering is partial, rather than total, because there may
exist elements x and y for which neither x R y nor y R x.

In domain theory, if D is a set of values including the
undefined value (bottom) then we can define a partial
ordering relation <= on D by

x <= y if x = bottom or x = y.

The constructed set D x D contains the very undefined element,
(bottom, bottom) and the not so undefined elements, (x,
bottom) and (bottom, x). The partial ordering on D x D is
then